Bistability and a differential equation model of the lac operon
Matthew Macauley
Department of Mathematical Sciences
Clemson University
http://www.math.clemson.edu/~macaule/
Math 4500, Spring 2016
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
1 / 23
Bistability
A system is bistable if it is capable of resting in two stable steady-states separated by
an unstable state.
From Wikipedia.
The threshold ODE: y 1
ry 1 My
1
y
T
.
In the threshold model for population growth, there are three steady-states,
0 T M:
carrying capacity (stable),
extinction threshold (unstable),
0 extinct (stable).
M
T
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
2 / 23
Types of bistability
For an example of bistability, consider the lac operon.
The expression level of the lac operon genes are either almost zero (“basal levels”),
or very high (thousands of times higher). There’s no “inbetween” state.
The precise expression level depends on the concentration level of intracellular
lactose. Let’s denote this parameter by p.
Now, let’s “tune” this parameter. The result might look like the graph on the left.
This is reversible bistability. In other situations, it may be irreversible (at right).
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
3 / 23
Hysteresis
In the case of reversible bistability, note that the up-threshold L2 of p is higher than
the down-threshold L1 of p.
This is hysteresis: a dependence of a state on its current state and past state.
Thermostat example
Consider a home thermostat set for 72 .
If the temperature is T
If the temperature is T
If 71 T
71, then the heat kicks on.
¡ 73, then the AC kicks on.
73, then we don’t know whether the heat or AC was on last.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
4 / 23
Hysteresis and the lac operon
If lactose levels are medium, then the state of the operon depends on whether or not
a cell was grown in a lactose-rich environment.
Lac operon example
Let rLs denote the concentration of intracellular lactose.
If rLs L1 , then the operon is OFF.
If rLs ¡ L2 , then the operon is ON.
rLs L2 , then the operon could be ON or OFF.
The region of bistability pL1 , L2 q has both induced and un-induced cells.
If L1
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
5 / 23
Hysteresis and the lac operon
The Boolean network models we’ve seen are too simple to capture bistability.
We’ll see two different ODE models of the lac operon that exhibit bistability.
These ODE models were designed using Michaelis–Menten equations from
mass-action kinetics which we learned about earlier.
In a later lecture, we’ll see how bistability can indeed be captured in a Boolean
network system.
In general, bistable systems tend to have positive feedback loops (in their “wiring
diagrams”) or double-negative feedback loops (=positive feedback).
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
6 / 23
Modeling dilution in protein concentration due to bacterial growth
E. coli grows fast! It can double in 20 minutes. Thus, reasonable ODE models
involving concentration shouldn’t assume that volume is constant.
Let’s define:
V
average volume of an E. coli bacterial cell.
number of molecules of protein X in that cell.
Let x
Assumptions about these derivatives:
cell volume increases exponentially in time:
degradation of X is exponential:
The concentration of x is rx s d rx s
dt
x 1V
V 1x
M. Macauley (Clemson)
1
V2
x
V
dx
dt
βx.
dV
dt
µV .
, and the derivative of this is (by the quotient rule):
βxV µVx
1
V2
β
Bistability & an ODE model of the lac operon
µ
x
V
pβ
µqrx s.
Math 4500, Spring 2016
7 / 23
Modeling of lactose repressor dynamics
Assumptions
Lac repressor protein is produced at a constant rate.
Laws of mass-action kinetics.
Repressor binds to allolactose:
R
Ýá
nA â
Ý RAn
K1
1
d rRAn s
dt
Assume the reaction is at equilibrium:
r
d RAn
dt
s
K1 rR srAsn rRAn s
0, and so K1 rRrRAsrAss .
n
n
The repressor protein binds to the operator region if there is no allolactose:
O
R
ÝKá
â
Ý OR
1
2
d rOR s
dt
Assume the reaction is at equilibrium:
M. Macauley (Clemson)
r s
d OR
dt
K2 rO srR s rOR s.
0, and so K2 rOrORsrRss .
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
8 / 23
Modeling of lactose repressor dynamics
total operator concentration (a constant). Then, using K2 rOrORsrRss ,
Otot rO s rOR s rO s K2 rO srR s rO sp1 K2 qrR s .
Therefore, OrO s 1 K1 rR s . “Proportion of free (unbounded) operator sites.”
Let Otot
tot
2
Let Rtot be total concentration of the repressor protein (constant):
rR s rOR s rRAn s
(
Assume only a few molecules of operator sites per cell: rOR s ! max rR s, rRAn s :
Rtot rR s rRAn s rR s K1 rR srAsn
Rtot
Eliminating rRAn s, we get rR s 1
Rtot
.
K1 rAsn
Now, the proportion of free operator sites is:
rO s Otot
1
where K
1
1
K2 rR s
1
p
1
K2 1 KRtot
n
1 A
rs
1
q1
K1 rAsn
K1 rAsn
K1 rAsn
,
K1 rAsn
loooooomoooooon
K1
prAsq
: f
K2 Rtot .
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
9 / 23
Modeling of lactose repressor dynamics
Summary
The proportion of free operator sites is
rO s 1
Otot
K
K1 rAsn
,
K1 rAsn
loooooomoooooon
where K
1
K2 Rtot .
prAsq
: f
Remarks
The function f prAsq is (almost) a Hill function of coefficient n.
f prAs 0q ¡ 0 “minimal basal level of gene expression.”
f is increasing in rAs, when rAs ¥ 0.
lim f prAsq 1 “with lots of allolactose, gene expression level is max’ed.”
rAsÑ8
M. Macauley (Clemson)
1
K
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
10 / 23
Modeling time-delays
The production of mRNA from DNA via transcription is not an instantaneous
process; suppose it takes time τ ¡ 0.
Thus, the production rate of mRNA is not a function of allolactose at time t, but
rather at time t τ .
Suppose protein P decays exponentially, and its concentration is p pt q.
dp
dt
µp ùñ
»t
t
τ
dp
p
µ
»t
t
τ
dt .
Integrating yields
t
p pt q
µt dt ln
µrt pt τ qs µτ .
p pt τ q
t τ
t τ
t
ln p pt q
Exponentiating both sides yields
pq
p q
p t
p t τ
e µτ ,and so
p pt q e µτ p pt
M. Macauley (Clemson)
τ q.
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
11 / 23
A 3-variable ODE model
Consider the following 3 quantities, which represent concentrations of:
M pt q mRNA,
B pt q β-galactosidase,
Apt q allolactose.
Assumption: Internal lactose (L) is available and is a parameter.
The model (Yildirim and Mackey, 2004)
dM
dt
dB
dt
dA
dt
αM K1
αB e µτ
K1 pe µτM AτM qn
K1 pe µτM AτM qn
B
MτB
γrM M
γrB B
αA B KL L L βA B KA A
A
γrA A
These are delay differential equations, with discrete time delays due to the
transcription and translation processes.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
12 / 23
3-variable ODE model
ODE for β-galactosidase (B)
dB
dt
αB e µτ
B
Mτ B
γrB B,
Justification:
γ
rB B γB B µB represents loss due to β-galactosidase degredation and
dilution from bacterial growth.
Production rate of β-galactosidase, is proportional to mRNA concentration.
τB time required for translation of β-galactosidase from mRNA, and
MτB : M pt τB q.
e µτB MτB accounts for the time-delay due to translation.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
13 / 23
3-variable ODE model
ODE for mRNA (M)
dM
dt
αM K1
K1 pe µτM AτM qn
K1 pe µτM AτM qn
γrM M
Justification:
rM M γM M µM represents loss due to mRNA degredation and dilution from
γ
bacterial growth.
Production rate of mRNA is proportional to fraction of free operator sites,
rO s 1
Otot
1
K1 rAsn
K1 rAsn
f prAsq.
The constant τM ¡ 0 represents the time-delay due to transcription of mRNA
from DNA. Define AτM : Apt τM q.
The term e µτM AτM accounts for the concentration of A at time t τM , and
dilution due to bacterial growth.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
14 / 23
3-variable ODE model
ODE for allolactose (A)
dA
dt
αA B KL L L βA B KA A
A
γrA A
Justification:
γ
rA A γA A µA represents loss due to allolactose degredation and dilution
from bacterial growth.
The first term models production of allolactose from the chemical reaction
β gal
lac ÝÑ allo.
The second term models loss of allolactose from the chemical reaction
β gal
allo ÝÑ glucose & galactose.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
15 / 23
A 3-variable ODE model
Steady-state analysis
To find the steady states, we must solve the nonlinear system of equations:
0 αM
1
K
K1 pe µτM AτM qn
K1 pe µτM AτM qn
γrM M
0 αB e µτB MτB
0 αA B
L
KL
γrB B
βA B KA A
L
A
γrA A
This was done by Yildirim et al. (2004). They set L 50 103 mM, which was in
the “bistable range.”
They also estimated the parameters through an extensive literature search.
Finally, they estimated µ 3.03 102 min1 by fitting the ODE models to
experimental data.
Steady states
I.
II.
III.
M. Macauley (Clemson)
A (mM)
4.27 103
1.16 102
6.47 102
M (mM)
4.57 107
1.38 106
3.28 105
Bistability & an ODE model of the lac operon
B (mM)
2.29 107
6.94 107
1.65 105
Math 4500, Spring 2016
16 / 23
3-variable ODE model
Figure: Bistability is L, A space. The y -axis is in logarithmic scale. For a range of L
concentrations there are three coexisting steady states for the allolactose concentration.
p
M. Macauley (Clemson)
q
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
17 / 23
3-variable ODE model
Figure: Time series simulations of mRNA, β-galactosidase and allolactose concentrations.
These were produced by numerically solving the 3-variable model using L 50 103 mM,
which is in the bistable region.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
18 / 23
5-variable ODE model
Consider the following 5 variables, which represent concentrations of:
M pt q mRNA,
B pt q β-galactosidase,
Apt q allolactose.
P pt q lac permease.
Lpt q intracellular lactose.
The model (Yildirim and Mackey, 2004)
dM
dt
dB
dt
dA
dt
dP
dt
dL
dt
αM K1
αB e µτ
K1 pe µτM AτM qn
K1 pe µτM AτM qn
B
rM M
Γ0 γ
γrB B
Mτ B
αA B KL L L βA B KA A
αP e µpτ
B
M. Macauley (Clemson)
q Mτ
τP
B
αL P KL Le
e
τP
Le
A
γrA A
γrP P
βL P KL L
e
e
L
αA B KL L L γrL L
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
19 / 23
5-variable ODE model
Remarks
The only difference in the ODE for M is the extra term Γ0 which describes the
basal transcription rate (in the absence of extracellular lactose).
The ODEs for B and A are the same as in the 3-variable model.
The ODE for P is very similar to the one for B:
production rate of lac permease is proportional to mRNA concentration, with a
time-delay.
the 2nd term accounts for loss due to degredation and dilution.
The ODE for lactose,
dL
dt
αL P KL Le
e
Le
βL P KL L
e
1
L
αA B KL L L γrL L,
is justified by the following:
The 1st term models gain due to transport of external lactose by lac permease.
The 2nd term accounts for loss due to this process being reversible.
ÝÑ
β gal
The 3rd term describes loss due to lac
allo.
the 4th term accounts for loss due to degredation and dilution.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
20 / 23
A 5-variable ODE model
To find the steady states, we set M 1
nonlinear system of equations.
A1 B 1 L1 P 1 0 and solve the resulting
This was done by Yildirim et al. (2004). They set Le
the “bistable range.”
50 103 mM, which was in
They also estimated the parameters through an extensive literature search.
Finally, they estimated µ 2.26 102 min1 by fitting the ODE models to
experimental data.
SS’s
I.
II.
III.
A (nM)
7.85 103
2.64 102
3.10 101
M. Macauley (Clemson)
M (mM)
2.48 106
7.58 106
5.80 104
B (mM)
1.68 106
5.13 106
3.92 104
L (mM)
1.69 101
2.06 101
2.30 101
Bistability & an ODE model of the lac operon
P (mM)
3.46 105
1.05 104
8.09 103
Math 4500, Spring 2016
21 / 23
5-variable ODE model
Figure: Bistability is L, A space. The y -axis is in logarithmic scale. For a range of Le
concentrations there are three coexisting steady states for the allolactose concentration.
p
M. Macauley (Clemson)
q
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
22 / 23
5-variable ODE model
Figure: Time series simulations of mRNA, β-galactosidase and allolactose concentrations.
These were produced by numerically solving the 3-variable model using Le
50 103 ,
which is in the bistable region.
M. Macauley (Clemson)
Bistability & an ODE model of the lac operon
Math 4500, Spring 2016
23 / 23